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Asymptotics of blowup solutions for the aggregation equation
1. | Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC V5A 1S6, Canada |
2. | 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095 |
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. |
[2] |
D. G. Aronson and J. L. Vázquez, Anomalous exponents in nonlinear diffusion, J. Nonlinear Sci., 5 (1995), 29-56. |
[3] |
D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Modél. Math. Anal. Numér., 31 (1997), 615-641. |
[4] |
A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 45-65. |
[5] |
A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.
doi: 10.1088/0951-7715/22/3/009. |
[6] |
A. L. Bertozzi, J. B. Garnett and T. Laurent, Characterization of radially symmetric finite time blowup in multidimensional aggregation equations, to appear in SIAM J. Math. Anal., 2012. |
[7] |
A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\mathbf R^n$, Comm. Math. Phys., 274 (2007), 717-735.
doi: 10.1007/s00220-007-0288-1. |
[8] |
A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Comm. Pur. Appl. Math., 64 (2011), 45-83.
doi: 10.1002/cpa.20334. |
[9] |
M. Bodnar and J. J. L. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differential Equations, 222 (2006), 341-380.
doi: 10.1016/j.jde.2005.07.025. |
[10] |
M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Anal. Real World Appl., 8 (2007), 939-958.
doi: 10.1016/j.nonrwa.2006.04.002. |
[11] |
M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Networks and Heterogeneous Media, 3 (2008), 749-785.
doi: 10.3934/nhm.2008.3.749. |
[12] |
E. Caglioti and C. Villani, Homogeneous cooling states are not always good approximations to granular flows, Arch. Ration. Mech. Anal., 163 (2002), 329-343.
doi: 10.1007/s002050200204. |
[13] |
J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.
doi: 10.1215/00127094-2010-211. |
[14] |
Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.
doi: 10.1016/j.physd.2007.05.007. |
[15] |
H. Dong, The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions, Communications in Mathematical Physics, 304 (2011), 649-664.
doi: 10.1007/s00220-011-1237-6. |
[16] |
M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and Lincoln S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse, Phys. Rev. Lett., 96 (2006), article 104302. |
[17] |
J. Eggers and M. A. Fontelos, The role of self-similarity in singularities of partial differential equations, Nonlinearity, 22 (2009), R1-R44.
doi: 10.1088/0951-7715/22/1/R01. |
[18] |
D. D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility, Phys. Rev. Lett., 95 (2005), article 226106.
doi: 10.1103/PhysRevLett.95.226106. |
[19] |
D. D. Holm and V. Putkaradze, Formation of clumps and patches in self-aggregation of finite-size particles, Phys. D, 220 (2006), 183-196.
doi: 10.1016/j.physd.2006.07.010. |
[20] |
Y. Huang, "Self-Similar Blowup Solutions of the Aggregation Equation,'' Ph.D thesis, University of California Los Angeles, 2010. Available online as UCLA CAM Report 10-66. |
[21] |
Y. Huang and A. L. Bertozzi, Self-similar blowup solutions to an aggregation equation in $R^n$, SIAM J. Appl. Math., 70 (2010), 2582-2603.
doi: 10.1137/090774495. |
[22] |
Y. Huang, T. Witelski and A. L. Bertozzi, Anomalous exponents of self-similar blowup solution to an aggregation equation in odd dimensions, preprint, 2010. |
[23] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[24] |
T. Laurent, Local and global existence for an aggregation equation, Comm. Partial Differential Equations, 32 (2007), 1941-1964. |
[25] |
H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal., 172 (2004), 407-428.
doi: 10.1007/s00205-004-0307-8. |
[26] |
A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.
doi: 10.1007/s002850050158. |
[27] |
A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation, J. Math. Biol., 47 (2003), 353-389.
doi: 10.1007/s00285-003-0209-7. |
[28] |
D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66.
doi: 10.1007/s00285-004-0279-1. |
[29] |
C. M. Topaz, A. J. Bernoff, S. Logan and W. Toolson, A model for rolling swarms of locusts, The European Physical Journal-Special Topics, 157 (2008), 93-109.
doi: 10.1140/epjst/e2008-00633-y. |
[30] |
C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174 (electronic).
doi: 10.1137/S0036139903437424. |
[31] |
C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
[32] |
G. Toscani, One-dimensional kinetic models of granular flows, M2AN Math. Model. Numer. Anal., 34 (2000), 1277-1291.
doi: 10.1051/m2an:2000127. |
[33] |
C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. |
show all references
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. |
[2] |
D. G. Aronson and J. L. Vázquez, Anomalous exponents in nonlinear diffusion, J. Nonlinear Sci., 5 (1995), 29-56. |
[3] |
D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Modél. Math. Anal. Numér., 31 (1997), 615-641. |
[4] |
A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 45-65. |
[5] |
A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.
doi: 10.1088/0951-7715/22/3/009. |
[6] |
A. L. Bertozzi, J. B. Garnett and T. Laurent, Characterization of radially symmetric finite time blowup in multidimensional aggregation equations, to appear in SIAM J. Math. Anal., 2012. |
[7] |
A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in $\mathbf R^n$, Comm. Math. Phys., 274 (2007), 717-735.
doi: 10.1007/s00220-007-0288-1. |
[8] |
A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Comm. Pur. Appl. Math., 64 (2011), 45-83.
doi: 10.1002/cpa.20334. |
[9] |
M. Bodnar and J. J. L. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differential Equations, 222 (2006), 341-380.
doi: 10.1016/j.jde.2005.07.025. |
[10] |
M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Anal. Real World Appl., 8 (2007), 939-958.
doi: 10.1016/j.nonrwa.2006.04.002. |
[11] |
M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Networks and Heterogeneous Media, 3 (2008), 749-785.
doi: 10.3934/nhm.2008.3.749. |
[12] |
E. Caglioti and C. Villani, Homogeneous cooling states are not always good approximations to granular flows, Arch. Ration. Mech. Anal., 163 (2002), 329-343.
doi: 10.1007/s002050200204. |
[13] |
J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.
doi: 10.1215/00127094-2010-211. |
[14] |
Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.
doi: 10.1016/j.physd.2007.05.007. |
[15] |
H. Dong, The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions, Communications in Mathematical Physics, 304 (2011), 649-664.
doi: 10.1007/s00220-011-1237-6. |
[16] |
M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and Lincoln S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse, Phys. Rev. Lett., 96 (2006), article 104302. |
[17] |
J. Eggers and M. A. Fontelos, The role of self-similarity in singularities of partial differential equations, Nonlinearity, 22 (2009), R1-R44.
doi: 10.1088/0951-7715/22/1/R01. |
[18] |
D. D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility, Phys. Rev. Lett., 95 (2005), article 226106.
doi: 10.1103/PhysRevLett.95.226106. |
[19] |
D. D. Holm and V. Putkaradze, Formation of clumps and patches in self-aggregation of finite-size particles, Phys. D, 220 (2006), 183-196.
doi: 10.1016/j.physd.2006.07.010. |
[20] |
Y. Huang, "Self-Similar Blowup Solutions of the Aggregation Equation,'' Ph.D thesis, University of California Los Angeles, 2010. Available online as UCLA CAM Report 10-66. |
[21] |
Y. Huang and A. L. Bertozzi, Self-similar blowup solutions to an aggregation equation in $R^n$, SIAM J. Appl. Math., 70 (2010), 2582-2603.
doi: 10.1137/090774495. |
[22] |
Y. Huang, T. Witelski and A. L. Bertozzi, Anomalous exponents of self-similar blowup solution to an aggregation equation in odd dimensions, preprint, 2010. |
[23] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[24] |
T. Laurent, Local and global existence for an aggregation equation, Comm. Partial Differential Equations, 32 (2007), 1941-1964. |
[25] |
H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal., 172 (2004), 407-428.
doi: 10.1007/s00205-004-0307-8. |
[26] |
A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.
doi: 10.1007/s002850050158. |
[27] |
A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation, J. Math. Biol., 47 (2003), 353-389.
doi: 10.1007/s00285-003-0209-7. |
[28] |
D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66.
doi: 10.1007/s00285-004-0279-1. |
[29] |
C. M. Topaz, A. J. Bernoff, S. Logan and W. Toolson, A model for rolling swarms of locusts, The European Physical Journal-Special Topics, 157 (2008), 93-109.
doi: 10.1140/epjst/e2008-00633-y. |
[30] |
C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174 (electronic).
doi: 10.1137/S0036139903437424. |
[31] |
C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
[32] |
G. Toscani, One-dimensional kinetic models of granular flows, M2AN Math. Model. Numer. Anal., 34 (2000), 1277-1291.
doi: 10.1051/m2an:2000127. |
[33] |
C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. |
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